x2 t08 04 inequality techniques (2012)
TRANSCRIPT
Types of Proof
Types of Proof 1. Deductive Proof
Start with known facts and deduce what you are trying to prove.
Types of Proof 1. Deductive Proof
Start with known facts and deduce what you are trying to prove.
2. Inductive Proof
Assume what you are trying to prove and induce a solution.
Types of Proof 1. Deductive Proof
Start with known facts and deduce what you are trying to prove.
2. Inductive Proof
Assume what you are trying to prove and induce a solution.
3. Proof by Contradiction
Assume the opposite of what you are trying to prove and create a
contradiction.
Contradiction means the original assumption is incorrect, therefore
the opposite must be true.
Inequality Techniques
Inequality Techniques 0 prove easier to becan it , prove To yxyx
Inequality Techniques 0 prove easier to becan it , prove To yxyx
2
Prove 1995 e.g.22 qp
pqi
Inequality Techniques 0 prove easier to becan it , prove To yxyx
2
Prove 1995 e.g.22 qp
pqi
pqqp
2
22
Inequality Techniques 0 prove easier to becan it , prove To yxyx
2
Prove 1995 e.g.22 qp
pqi
pqqp
2
22
2
2 22 qpqp
Inequality Techniques 0 prove easier to becan it , prove To yxyx
2
Prove 1995 e.g.22 qp
pqi
pqqp
2
22
2
2 22 qpqp
2
2qp
0
Inequality Techniques 0 prove easier to becan it , prove To yxyx
2
Prove 1995 e.g.22 qp
pqi
pqqp
2
22
2
2 22 qpqp
2
2qp
0
pqqp
2
22
Inequality Techniques 0 prove easier to becan it , prove To yxyx
2
Prove 1995 e.g.22 qp
pqi
pqqp
2
22
2
2 22 qpqp
2
2qp
0
pqqp
2
22
OR 2 2
Assume 2
p qpq
Inequality Techniques 0 prove easier to becan it , prove To yxyx
2
Prove 1995 e.g.22 qp
pqi
pqqp
2
22
2
2 22 qpqp
2
2qp
0
pqqp
2
22
OR 2 2
Assume 2
p qpq
2 22pq p q
2 2
2
0 2
0 ( )
p pq q
p q
Inequality Techniques 0 prove easier to becan it , prove To yxyx
2
Prove 1995 e.g.22 qp
pqi
pqqp
2
22
2
2 22 qpqp
2
2qp
0
pqqp
2
22
OR 2 2
Assume 2
p qpq
2 22pq p q
2 2
2
0 2
0 ( )
p pq q
p q
2But ( ) 0p q
Inequality Techniques 0 prove easier to becan it , prove To yxyx
2
Prove 1995 e.g.22 qp
pqi
pqqp
2
22
2
2 22 qpqp
2
2qp
0
pqqp
2
22
OR 2 2
Assume 2
p qpq
2 22pq p q
2 2
2
0 2
0 ( )
p pq q
p q
2But ( ) 0p q 2 2
2
p qpq
acbcabcbaii 222 Prove a) 1994
acbcabcbaii 222 Prove a) 1994
02ba
acbcabcbaii 222 Prove a) 1994
02ba
02 22 baba
acbcabcbaii 222 Prove a) 1994
02ba
02 22 baba
abba 222
acbcabcbaii 222 Prove a) 1994
02ba
02 22 baba
abba 222
bccb
acca
2
2
22
22
acbcabcbaii 222 Prove a) 1994
02ba
02 22 baba
abba 222
bccb
acca
2
2
22
22
bcacabcba 222222 222
acbcabcbaii 222 Prove a) 1994
02ba
02 22 baba
abba 222
bccb
acca
2
2
22
22
bcacabcba 222222 222
bcacabcba 222
acbcabcbaii 222 Prove a) 1994
02ba
02 22 baba
abba 222
bccb
acca
2
2
22
22
bcacabcba 222222 222
bcacabcba 222
3
1 prove ,1 If b) bcacabcba
acbcabcbaii 222 Prove a) 1994
02ba
02 22 baba
abba 222
bccb
acca
2
2
22
22
bcacabcba 222222 222
bcacabcba 222
3
1 prove ,1 If b) bcacabcba
bcacabcbacba 22222
acbcabcbaii 222 Prove a) 1994
02ba
02 22 baba
abba 222
bccb
acca
2
2
22
22
bcacabcba 222222 222
bcacabcba 222
3
1 prove ,1 If b) bcacabcba
bcacabcbacba 22222
bcacabbcacabcba 22
acbcabcbaii 222 Prove a) 1994
02ba
02 22 baba
abba 222
bccb
acca
2
2
22
22
bcacabcba 222222 222
bcacabcba 222
3
1 prove ,1 If b) bcacabcba
bcacabcbacba 22222
bcacabbcacabcba 22
23 cbabcacab
acbcabcbaii 222 Prove a) 1994
02ba
02 22 baba
abba 222
bccb
acca
2
2
22
22
bcacabcba 222222 222
bcacabcba 222
3
1 prove ,1 If b) bcacabcba
bcacabcbacba 22222
bcacabbcacabcba 22
23 cbabcacab
3
1
13
bcacab
bcacab
3
3
1 Prove c) abccba
3
3
1 Prove c) abccba
bcacabcba 222
3
3
1 Prove c) abccba
bcacabcba 222
0222 bcacabcba
3
3
1 Prove c) abccba
bcacabcba 222
0222 bcacabcba
0222 bcacabcbacba
3
3
1 Prove c) abccba
bcacabcba 222
0222 bcacabcba
0222 bcacabcbacba
022322
2223222223
bcacabcccbca
cbabcabbcbbaabccabaacaba
3
3
1 Prove c) abccba
bcacabcba 222
0222 bcacabcba
0222 bcacabcbacba
022322
2223222223
bcacabcccbca
cbabcabbcbbaabccabaacaba
03333 abccba
3
3
1 Prove c) abccba
bcacabcba 222
0222 bcacabcba
0222 bcacabcbacba
022322
2223222223
bcacabcccbca
cbabcabbcbbaabccabaacaba
03333 abccba
abccba 333
3
1
3
3
1 Prove c) abccba
bcacabcba 222
0222 bcacabcba
0222 bcacabcbacba
022322
2223222223
bcacabcccbca
cbabcabbcbbaabccabaacaba
03333 abccba
abccba 333
3
1
3
1
3
1
3
1
,,let ccbbaa
3
3
1 Prove c) abccba
bcacabcba 222
0222 bcacabcba
0222 bcacabcbacba
022322
2223222223
bcacabcccbca
cbabcabbcbbaabccabaacaba
03333 abccba
abccba 333
3
1
3
1
3
1
3
1
,,let ccbbaa
3
1
3
1
3
1
3
1cbacba
3
3
1 Prove c) abccba
bcacabcba 222
0222 bcacabcba
0222 bcacabcbacba
022322
2223222223
bcacabcccbca
cbabcabbcbbaabccabaacaba
03333 abccba
abccba 333
3
1
3
1
3
1
3
1
,,let ccbbaa
3
1
3
1
3
1
3
1cbacba
3
3
1abccba
nn
n aaan
aaa
21
21
Mean GeometricMean Arithmetic
nn
n aaan
aaa
21
21
Mean GeometricMean Arithmetic
1 prove ,8111 Suppose d) xyzzyx
nn
n aaan
aaa
21
21
Mean GeometricMean Arithmetic
1 prove ,8111 Suppose d) xyzzyx
81
8111
xyzyzxzzxyyx
zyx
nn
n aaan
aaa
21
21
Mean GeometricMean Arithmetic
1 prove ,8111 Suppose d) xyzzyx
81
8111
xyzyzxzzxyyx
zyx
3
3
1xyzzyx AM GM
nn
n aaan
aaa
21
21
Mean GeometricMean Arithmetic
1 prove ,8111 Suppose d) xyzzyx
81
8111
xyzyzxzzxyyx
zyx
3
3
1xyzzyx
33 xyzzyx
AM GM
nn
n aaan
aaa
21
21
Mean GeometricMean Arithmetic
1 prove ,8111 Suppose d) xyzzyx
81
8111
xyzyzxzzxyyx
zyx
3
3
1xyzzyx
33 xyzzyx
33 xzyzxyxzyzxy
AM GM
nn
n aaan
aaa
21
21
Mean GeometricMean Arithmetic
1 prove ,8111 Suppose d) xyzzyx
81
8111
xyzyzxzzxyyx
zyx
3
3
1xyzzyx
33 xyzzyx
33 xzyzxyxzyzxy
3 2223 zyxxzyzxy
AM GM
nn
n aaan
aaa
21
21
Mean GeometricMean Arithmetic
1 prove ,8111 Suppose d) xyzzyx
81
8111
xyzyzxzzxyyx
zyx
3
3
1xyzzyx
33 xyzzyx
33 xzyzxyxzyzxy
3 2223 zyxxzyzxy
233 xyzxzyzxy
AM GM
81 xyzyzxzxyzyx
81 xyzyzxzxyzyx
83312
33 xyzxyzxyz
81 xyzyzxzxyzyx
83312
33 xyzxyzxyz
83313
32
33 xyzxyzxyz
81 xyzyzxzxyzyx
83312
33 xyzxyzxyz
83313
32
33 xyzxyzxyz
813
3 xyz
81 xyzyzxzxyzyx
83312
33 xyzxyzxyz
83313
32
33 xyzxyzxyz
813
3 xyz
21 3 xyz
81 xyzyzxzxyzyx
83312
33 xyzxyzxyz
83313
32
33 xyzxyzxyz
813
3 xyz
21 3 xyz
13 xyz
81 xyzyzxzxyzyx
83312
33 xyzxyzxyz
83313
32
33 xyzxyzxyz
813
3 xyz
21 3 xyz
13 xyz
1xyz
OR 1 2x x AM GM
OR 1 2x x
1 2y y
1 2z z
AM GM
OR 1 2x x
1 2y y
1 2z z
(1 )(1 ) 1 2 2 2
8
x y z x y z
xyz
AM GM
OR 1 2x x
1 2y y
1 2z z
(1 )(1 ) 1 2 2 2
8
x y z x y z
xyz
8 8 xyz
AM GM
OR 1 2x x
1 2y y
1 2z z
(1 )(1 ) 1 2 2 2
8
x y z x y z
xyz
8 8 xyz
1 1
1
1
xyz
xyz
xyz
AM GM
9 2 2 2 1 1 1
Prove iiia b c a b b c a c a b c
9 2 2 2 1 1 1
Prove iiia b c a b b c a c a b c
1 1 1
2 2
4
a b aba b ab
9 2 2 2 1 1 1
Prove iiia b c a b b c a c a b c
1 1 1
2 2
4
a b aba b ab
1 1 4
a b a b
9 2 2 2 1 1 1
Prove iiia b c a b b c a c a b c
1 1 1
2 2
4
a b aba b ab
1 1 4
a b a b
1 1 4
1 1 4
b c b c
a c a c
9 2 2 2 1 1 1
Prove iiia b c a b b c a c a b c
1 1 1
2 2
4
a b aba b ab
1 1 4
a b a b
1 1 4
1 1 4
b c b c
a c a c
2 2 2 4 4 4
a b c a b b c a c
9 2 2 2 1 1 1
Prove iiia b c a b b c a c a b c
1 1 1
2 2
4
a b aba b ab
1 1 4
a b a b
1 1 4
1 1 4
b c b c
a c a c
2 2 2 4 4 4
a b c a b b c a c
1 1 1 2 2 2
a b c a b b c a c
9 2 2 2 1 1 1
Prove iiia b c a b b c a c a b c
1 1 1
2 2
4
a b aba b ab
1 1 4
a b a b
1 1 4
1 1 4
b c b c
a c a c
2 2 2 4 4 4
a b c a b b c a c
1 1 1 2 2 2
a b c a b b c a c
3 31 1 1 1
3 3
9
a b c abca b c abc
9 2 2 2 1 1 1
Prove iiia b c a b b c a c a b c
1 1 1
2 2
4
a b aba b ab
1 1 4
a b a b
1 1 4
1 1 4
b c b c
a c a c
2 2 2 4 4 4
a b c a b b c a c
1 1 1 2 2 2
a b c a b b c a c
3 31 1 1 1
3 3
9
a b c abca b c abc
1 1 1 9
a b c a b c
9 2 2 2 1 1 1
Prove iiia b c a b b c a c a b c
1 1 1
2 2
4
a b aba b ab
1 1 4
a b a b
1 1 4
1 1 4
b c b c
a c a c
2 2 2 4 4 4
a b c a b b c a c
1 1 1 2 2 2
a b c a b b c a c
3 31 1 1 1
3 3
9
a b c abca b c abc
1 1 1 9
a b c a b c
1 1 1 9
2
2 2 2 9
a b b c a c a b c
a b b c a c a b c
9 2 2 2 1 1 1
Prove iiia b c a b b c a c a b c
1 1 1
2 2
4
a b aba b ab
1 1 4
a b a b
1 1 4
1 1 4
b c b c
a c a c
2 2 2 4 4 4
a b c a b b c a c
1 1 1 2 2 2
a b c a b b c a c
3 31 1 1 1
3 3
9
a b c abca b c abc
1 1 1 9
a b c a b c
1 1 1 9
2
2 2 2 9
a b b c a c a b c
a b b c a c a b c
9 2 2 2 1 1 1
a b c a b b c a c a b c
9 2 2 2 1 1 1
Prove iiia b c a b b c a c a b c
1 1 1
2 2
4
a b aba b ab
1 1 4
a b a b
1 1 4
1 1 4
b c b c
a c a c
2 2 2 4 4 4
a b c a b b c a c
1 1 1 2 2 2
a b c a b b c a c
3 31 1 1 1
3 3
9
a b c abca b c abc
1 1 1 9
a b c a b c
1 1 1 9
2
2 2 2 9
a b b c a c a b c
a b b c a c a b c
9 2 2 2 1 1 1
a b c a b b c a c a b c
Inequalities Sheet
Exercise 10D
Note: Cambridge 8H (Book 1); 28